In its main body, the book develops the mathematical tools necessary in Relativity. From a physicist’s point of view, this is done in an adequate, not too technical way, using a language that is understandable for both mathematicians and physicists. Most essential topics (differential manifolds, tensors, spinors, forms, tetrads,...) are covered. In the Relativity part, only some selected important topics are treated, which - besides the fundamentals on field equations and physics and measurements in a curved background - are the spherically symmetric fields (including a few pages on cosmology) and the Kerr metric.
When browsing through the book, one will find besides the essentials many well presented details usually not treated, or treated differently, in comparable books. Among them are the connector (for transport along a given curve), the revival of Synge’s world function, the variational principle for a perfect fluid, the centre of mass’s equation of motion, and the superradiance in the Kerr metric.
The book is carefully written and printed but amusingly misprints are accumulated in the quotations with German titles (11 titles with 14 misprints), among them (and also in the text) the misdating of Hilbert’s work on the action of Einstein’s theory into the year 1917 instead of 1915.
Judging by contents and presentation, I would locate this book between a monograph and a textbook. It does not quite meet the requirements for a monograph on the mathematics used in relativity because a few topics such as the Maurer-Cartan equation, the integral theorems and the theory of integration, Lie groups, non-orthogonal and null tetrads, and the Newman- Penrose formalism are missing or only mentioned in passing. It is not quite a textbook since on several occasions a motivation for what is going on is missing or - in particular in the more physically oriented second part - results are simply given without going into the details. The book is just the authors’ view of what they feel is necessary and worthwile in this field, and because of its many novel features in the details and its carefully spelt out applications it deserves the attention and the interest of scientists and graduate students working on the mathematical or physical side of General Relativity.